Edit: I've thought about this more and realized that it's not as complicated as it seemed.
The object pronouns for the imperative are mostly the same as the ones you usually use for verbs; moi and toi are the only exceptions.
Chart & examples
Here are the object pronouns when used in the imperative:
Here are some examples of the imperative pronouns in action:
Dites-moi ce que vous voudriez. ← dire à moi
Montre-lui ta solution. ← montrer à lui
Parlons-leur de ce problème. ← parler à eux
C'est ton anniversaire. Fête-le ! ← fêter l'anniversaire
Ces biscuits sont-ils les vôtres ? Mangez-les donc ! ← manger les biscuits
Les informations qu'elles ont demandées ? Donne-les-leur. ← donner les informations à elles
Why would this be?
I find that these pronouns are best viewed as separate systems with a lot of overlap. When comparing that chart with this one, you'll see that all the pronouns cross rows (used for multiple persons) and columns (used in multiple scenarios). There's no hard-and-fast rule to predict which form a pronoun will take anywhere; you just have to know these charts.
However, Gilles' comment and answer suggest that this has to do with the position of the pronoun. If it comes before the verb, we use the weak form: me and te. But if it comes after the verb, we use the strong form.
We can test this to some degree using the negative imperative. The negative imperative generally has the pronoun before the verb. Sure enough, we get the normal object pronouns:
Ne me dites pas ce que vous voudriez. ← dire à moi
Ne t'en fais pas. ← faire à toi-même
But there's a colloquial version where you can put it after, and here it becomes the "strong" version:
Fais-toi-en pas. ← faire à toi-même (Daniel Lavoie, « Ça c'est ça »)
So the theory works — but hang on, which system is it that gives us the "strong" version?
- If we say it's the tonique, we have to explain why we use lui instead of elle.
- If we say it's the COI, we have to explain why we use moi and toi instead of me and te.
- Maybe it's the COI in general, but since me and te are weak forms, the tonique replaces them.
In any case, you can see my point: somewhere it becomes hard to explain why this or that system is used in a specific scenario,1 and you more or less have to just view it as a chart to memorize. That's why I made the hopefully handy ones above. :)
1 Without diving into historical linguistics to trace all the stages of each pronoun.